Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. II. Spin-singlet (excited) and spin-triplet (lowest) states of the Helium sequence

Abstract

As a continuation of Part I \cite{Part-1:2020} (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z\le 20$, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for the energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state $2^{1}S$ and the lowest spin-triplet $1^{3}S$ state. For both states it provides absolute accuracy for energy $\sim 10^{-3}$\,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $\sim 10^{-2}$. The Bressanini-Reynolds observation about the special form of the nodal surface of the $2^{1}S$ state of Helium is confirmed and extended to He-like ions with $Z>2$. Critical charges $Z=Z_B$, where ultra-compact trial functions lose their square-integrability, are estimated: $Z_B(1^{1}S)\approx Z_B(2^{1}S)\sim 0.905$ and $Z_B(1^{3}S)\sim 0.902$. For both states the Majorana formula - the energy as a second degree polynomial in $Z$ - provides accurately 4-5 significant digits for $Z\le 20$.

27 pages, 6 figures, 4 tables: typos fixed, 8 extra references added, the text essentially extended, to be published at Int Journ Quantum Chemistry

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